TABLE OF CONTENTS
\cftbeforechapskip\cftbeforechapskip

1pt

FLORIDA INTERNATIONAL UNIVERSITY

Miami, Florida

MULTINUCLEON SHORT-RANGE CORRELATION MODEL FOR NUCLEAR SPECTRAL FUNCTIONS

A dissertation submitted in partial fulfillment of the

requirements of the degree of

DOCTOR OF PHILOSOPHY

in

PHYSICS

by

Oswaldo Artiles

2017

To: Dean Michael R. Heithaus
College of Arts, Sciences and Education

This dissertation, written by Oswaldo Artiles, and entitled Multinucleon Short-range Correlation Model for Nuclear Spectral Functions, having been approved in respect to style and intellectual content, is referred to you for judgment.

We have read this dissertation and recommend that it be approved.

DEDICATION

{CJK}

UTF8ipxga

This dissertation is dedicated to Rita Elisa and Ana Rita, my grandmother and my mother, who gave me love and cared for me everyday of my childhood, and who taught me the values of honesty, respect for life, and love for hard work, values which have guided me during my entire life.

ACKNOWLEDGMENTS

I would like to acknowledge all the teachers and professors that help me and taught me during all years of my life as a student.

I am indebted to the members of my dissertation committee: Oren W. Maxwell, Werner Boeglin, and Gueo Grantcharov, who helped me during all my research. My special acknowledgment to my adviser, Misak Sargsian, who taught me a lot of physics and was always patient with my ignorance.

I wish to acknowledge the graduate program directors, Brian Raue, Rajamani Narayanan, and Jorge Rodriguez, as well the administrative staff of the the FIU physics department, specially Elizabeth Bergano-Smith, Omar Tolbert, Maria Martinez, John Omara, Ofelia Adan-Fernandez, and Robert Brown who support me with all their administrative and technical expertise.

I also thank Frank Vera, Dhiraj Maheswari, and Christopher Leon, my fellow PhD students of the group of nuclear theoretical physics, for the very helpful physics and mathematics discussions that I have had with them.

Finally, my greatest debt to my wife Ivia, for giving me love and for taking care of me with total dedication, and to my daughters Livia and Claudia and all my grandchildren: Mauricio, Ana, Isabella e Ignacio, for all their love and for giving me the most beautiful reasons to live.

This research would not have been possible without financial support. I would like to acknowledge the Department of Energy for providing the grant that supported my research assistantship.

ABSTRACT OF THE DISSERTATION

MULTINUCLEON SHORT-RANGE CORRELATION MODEL FOR NUCLEAR SPECTRAL FUNCTIONS

by

Oswaldo Artiles

Florida International University, 2017

Miami, Florida

Professor Misak M. Sargsian, Major Professor

The main goal of the research presented in my dissertation was to develop a theoretical model for relativistic nuclear spectral functions at high missing momenta and removal energies based on the multi-nucleon short-range correlation (SRC) model. The nuclear spectral functions are necessary for the description of high energy nuclear processes currently being studied at different labs such as JLAB, LHC and FNAL.

The model followed the effective Feynman diagrammatic approach in order to account for the relativistic effects important in the SRC domain. In addition to the two-nucleon (2N) SRC with center of mass motion contribution, the contribution of the three-nucleon SRCs to the spectral functions was also derived. The latter was modeled based on the assumption that the 3N SRCs are a product of two sequential short range nucleon-nucleon (NN) interactions.

The nuclear spectral functions models were derived from two theoretical frameworks for evaluating covariant Feynman diagrams: In the first, referred to as the virtual nucleon approximation, the Feynman diagrams were reduced to the time ordered non-covariant diagrams by evaluating the nucleon spectators in the SRC at their positive energy poles, neglecting explicitly the contribution from vacuum diagrams. In the second approach, referred to as the light-front approximation, the boost invariant nuclear spectral function was formulated in the light-front reference frame in which case the vacuum diagrams are kinematically suppressed and the bound nucleon is described by its light-front variables such as momentum fraction, transverse momentum and invariant mass.

On the basis of the derived nuclear spectral functions, the corresponding computational models were developed from which the numerical estimates of the SRC spectral functions, the SRC momentum distributions, and the SRC density matrices were obtained.

Table of Contents
List of Figures

Chapter 1 Introduction

Scattering experiments have been the most important experimental tools to reveal the structure of visible matter, specially the structure of atoms and nucleus. In 1911, Rutherford, through scattering experiments of alpha particles off atomic targets, discovered the atomic nucleus [1]. Experiments of disintegration of nitrogen nuclei in 1919, led Rutherford to conclude that the nucleus was a composite system of particles held together by a strong force [2]. Rutherford also gave the name of proton to the hydrogen nucleus, and predicted the neutron in 1920 [3]. The prediction of Rutherford was confirmed by Chadwick [5] and Curie-Joliot [6] who, independently, showed the existence of the neutron in 1932. The atomic nucleus has since been considered a system composed of strong interacting nucleons 111a common name for protons and neutrons [7], held together by strong short-range forces, of a different nature than those of the electromagnetic and gravitational forces. Since the atomic nucleus is the basic component of all the visible matter in the universe, a continuous and very important experimental and theoretical efforts have been ongoing since 1932 to understand and explain the nuclear and the nucleon structures at the fundamental level.

In 1935, Yukawa described the interaction of nucleons in the nucleus by means of a field of force associated with a particle or a quantum which was a carrier of this interaction [8]. Two particles, the pion+ and the pion-, were discovered in 1947 [9], and the neutral pion was isolated in 1950, completing a trilogy of particles that were identified as the carrier particles predicted by Yukawa.

The availability of high energy particle accelerators in the middle fifties and early sixties, allowed the discovery of a multitude of strong interacting particles collectively known as hadrons, that formed groups of particles with similar properties [55]. The first formal group classification of the hadrons was given by Gell Man [16] and independently by Yuval Ne’Man [17] in the so-called "Eightfold Way", that classified hadrons into subgroups identified with octet representations of the SU(3) group. The classification of hadrons proposed by Gell Man was confirmed by the 1964 discovery of the omega-minus particle, predicted in 1962. After this discovery Gell-Mann [20] and George Zweig [21, 22] independently predicted that hadrons were composed of elementary fermions named as quarks [54].

The ever increasing energy of modern particle accelerators and the complete knowledge of the electromagnetic interaction between electrons and nucleons or nuclei were essential to prove the existence of the quarks as fundamental components of hadrons in general, and nucleons in particular. In 1968, the first results from deep inelastic scattering experiments demonstrated that protons and neutrons are composite structures made up of particles with fractional electric charge as well as neutral particles [56, 57]. The charged particles were later identified with quarks predicted by Gell-Mann [20] and Zweig [21, 22], and the neutral particles with the massless gluons, the carriers of the strong interaction between quarks in hadrons.

The above mentioned experimental and theoretical efforts resulted in the formalization and acceptance of the quantum field theory of strong interactions: quantum chromodynamics (QCD), within QCD the vast majority of physical hadrons are many-body, highly relativistic systems composed of light quarks and massless gluons [54, 85]. Quantum Chromodynamics (QCD) is a non-abelian gauge field theory (Yang and Mills theory [13] ) that describes the strong interactions of colored quarks and gluons fields. Quantum Chromodynamics is the SU(3) component of the SU(3)xSU(2)xU(1) of the Standard Model of Particle Physics.

Quarks are strongly interacting fermions with spin 1/2 and, by convention positive parity. The electric charges of the quarks are -1/3 and 2/3. There are six different types of flavors of quarks: up (u), down (d), strange (s), charm (c), top (t) and bottom (b). In order to have antisymmetric wave functions (Pauli exclusion principle), the quarks carry color charges, which for convention are called: red (R), green (G) and blue(B). The color of quarks is a quantum number that was first proposed by Fritzsch and Gell-Man [31] who identified the extra symmetry of QCD with the color symmetry. As a result of the color charges, quarks are said to be in the fundamental representation of the SU(3) color Lie group.

Gluons are massless gauge bosons, with spin 1 and two polarization states, which mediate color interactions among quarks. They, represented through non-Abelian gauge fields in QCD, are responsible for binding the quarks together. Gluons carry color charges and hence interact with each other even in the absence of quarks [40]. Gluons transform under the adjoint representation of the Lie group SU(3) [117].

Even though hadrons are composite systems of quarks and gluons, the understanding of the nuclear structure is mainly based on the nucleonic degrees of freedom. Hence, a nucleus of mass number and atomic number , is commonly represented as a system of A nucleons, of which are protons and neutrons. The main picture of nuclei as a composite system of nucleons is taken from the mean field approximation, on which each nucleon is modeled as an independent particle moving in the mean field potential of the remaining nucleons [80]. However, the picture breaks down when the momentum of the bound nucleon exceeds 250 MeV/, where is the characteristic Fermi momentum of the nucleus described as a degenerate Fermi gas [28].

One of the major issues of modern nuclear physics is to understand the nuclear structure above the domain. The issue is directly related to understanding the nucleon-nucleon interaction at short distances inside the nucleus. The interaction prevents two nucleons from coming very close together demanding the existence of high momentum components in the nuclear ground state wave function. The close nucleon-nucleon interactions cannot be described in the context of nuclear mean field models and are commonly called multi-nucleon short-range correlations (SRC) [99].

Deep inelastic scattering (DIS) of leptons off the nucleus is one of the methods for probing nuclear structures at small distance scales. Before 1983, it was predicted that the quarks inside the nucleus followed the rules of the mean field model. According to which the quark-gluon dynamics in each nucleon was not influenced by the nuclear mean field. The fact that the quark momentum distributions for individual nucleons are modified in the nuclear medium was first observed by the European Muon Collaboration (EMC) experiment and is usually known as the EMC effect [38]. The discovery of the EMC effect is considered the starting event of a new era in nuclear QCD.

The importance of the high momentum properties of bound nucleon for nuclear EMC effects follows from the recent observations of apparent correlation between the medium modification of the quark momentum distributions and the strength of the two-nucleon short range correlations (SRCs) in nuclei[97, 100]. In order to improve our knowledge of the role of the QCD interactions in the nuclear dynamics, it is crucial to understand the role of the SRCs in the EMC effects [75].

As a result of the QCD evolution of nuclear parton distributions functions(PDFs)( see Eq. (1.3)), it is expected that at very large the knowledge of the high momentum component of the bound nucleons becomes important because of the contribution of quarks with momentum fractions () larger than the ones provided by an isolated nucleon (i.e. partons with )[125, 126].

The same is true for the reliable interpretation of neutrino-nuclei scattering experiments in which case both the medium modification of PDFs as well as the realistic treatment of SRCs are essential. One of the main results of the experiments is the discrepancy found between measurements of ( is the weak mixing angle) involving free particles and those involving bound nucleons in the nuclear medium, the so-called NuTev anomaly [77, 72, 73, 102]. A possible explanation to the corresponding results is derived from the presence of more energetics protons in neutron rich, large A, asymmetric nuclei which implies that u-quarks are more modified than d-quarks, resulting in a negative correction for the experimental value of for bound nucleons [111].

All the above described areas of research require models to predict the high momenta and binding energies of bound nucleons in the nuclei. Such models are described through the nuclear spectral functions that define the joint probability of finding a nucleon in the nucleus with momentum and removal (binding) energy . With the advent of the Large Hadron Collider and expected construction of electron-ion colliders as well as several ongoing neutrino-nuclei experiments, the knowledge of such spectral functions will be an important part of the theoretical interpretation of the data involving nuclear targets.

The present dissertation is divided in four major chapters. In chapter 2, the theoretical framework for the multinucleon short-range correlation model for nuclear spectral functions is developed. The diagrammatic method of calculation of the spectral function is described, as well as the mathematical models for the spectral function in two approaches: Virtual nucleon and light front approximations. From the mathematical models of chapter 2, computational models to obtain numerical estimates for the spectral functions, density matrices, and nucleon momentum distributions are developed in chapter 3. The numerical estimates of the spectral functions, density matrices, and nucleon momentum distributions are presented in chapter 4. Finally, the main conclusions and an outline of future projects are included in chapter 5.

In the rest of chapter 1, the relationship between lepton-nucleus scattering cross section and the nuclear spectral function is presented. Then, a brief description of the EMC effect and its relation to 2N SRC is included. The assumptions for modeling two and three nucleon short-range correlations are also described. Finally, some definitions and the range of validity of the nuclear spectral functions, including the treatment of the relativistic effects and the model for 2N SRC center of mass motion are presented.

1.1 Lepton-nucleus inclusive scattering cross section and nuclear spectral function

In the lepton-nucleus inclusive scattering reaction (), shown in Fig.1.1(a), X is the hadron final state and (A-1) the recoil system of bound nucleons, and only the final lepton is detected.

Figure 1.1: Lepton-nucleus inclusive scattering.

In the one-boson exchange approximation, if l and l’ are the same leptons ( electrons or muons) the reaction is mediated by either the weak neutral gauge boson, , or the photon . If l and l’ differs by one unit of charge, for instance (muon, neutrino) or (anti-muon, anti-neutrino), the reaction is mediated by the weak charged gauge bosons, or .

The kinematic variables for the lepton-nucleus inclusive scattering in the lab frame (nucleus rest frame) are: , for the four-momentum vector of the nucleus of mass ; and , for the four-momentum vectors of the initial l and final leptons, respectively. , is the four-momentum vector of the gauge boson, with defined as the virtuality of the gauge boson. If , where is the mass of the Z () gauge boson, and l=l’ (muon or electron) the reaction is defined by one-photon exchange.

Theoretical studies demonstrated that If , where is the mass of a nucleon. the size of the probe ( ) is smaller than the size of the nucleon, hence it can be considered that the lepton scatters off a bound nucleon inside the nucleus (see Fig.1.1(b) and (c)). In Fig 1.1, it is assumed that the final state interactions of the hadron final state () can be neglected. Such an assumption is referred to as the plane wave impulse approximation (PWIA) [82, 39]. The PWIA is justified for inclusive processes with large () such as deep inelastic scattering [113].

The total energy of the lepton-nucleon scattering (Fig. 1.1(b)) is defined as . If then the hadron final state X is a nucleon and the corresponding lepton-nucleus scattering is defined as quasi-elastic. If , the hadron final state X may consist of a nucleon and a pion () or baryonic excited resonances. In the region of deep inelastic scattering (DIS), 2.5 GeV and 2.0 GeV, the target nucleon breaks down in a final state X consisting of a collection of hadrons.

Deep inelastic lepton-nucleon scattering (DIS) has been fundamental in unveiling the structure of nucleons. In the parton model with very large Q [26, 29], the virtual photon interacts with one of the quarks of the bound nucleon (Fig 1.1(c)). Such a picture is relevant in the infinite momentum frame (IMF) of the nucleon. The lepton-nucleon center of mass frame is a good approximation to such IMF, on which the quarks (partons) within the nucleons are slowed down by Lorentz time dilation effects. In the IMF, therefore, the struck quark can be considered free, and characterized by the momentum fraction of the fast nucleon defined as the invariant Bjorken scaling variable , (0 1), [26].

The one-photon exchange differential cross section for unpolarized lepton-nucleon inclusive inelastic scattering can be expressed through two independent nucleon structure functions and as [11, 23, 24, 101]

(1.1)

where , and is the Mott cross section for electron-point charge scattering [4] , and the scattering angle. In the elastic scattering kinematics, and can be expressed as functions of the so called electric () and magnetic () form factors of the nucleon[15, 18, 35, 40].

In the inelastic scattering kinematics, and are functions of the two independent variables and . However within the partonic model of the nucleon, it was predicted that in deeps inelastic scattering (large ):

(1.2)

where the dimensionless nucleon structure functions, are independent of , that is independent of any mass scale (scale invariant), signaling the presence of free point-like quarks in the nucleons by satisfying the Bjorken scaling property[26].
The nucleon structure functions contain the information about the parton’s (quark’s) momentum distribution in the nucleon, namely

(1.3)

where is the quark electric charge, and is the parton longitudinal momentum distribution function in the infinite momentum frame, that is the probability that the struck quark carries a fraction of the nucleon momentum . Hence, the nucleon structure function gives the weighted, by the square of the parton electric charge, probability of finding a parton in the nucleon that carries a fraction of the total nucleon momentum.
The parton momentum distribution functions are normalized as:

(1.4)

where sums over all the partons, not just the charged ones that interact with the photon [34, 40, 108].

Figure 1.2: Lepton-nucleus deep inelastic inclusive scattering regions

Similar to the free nucleon case, the inclusive cross section for lepton-nucleus scattering can be expressed as follows

(1.5)

where and are the nuclear structure functions, that, within the above discussed plane wave impulse approximation (PWIA), can be expressed as a function of the nucleon structure functions, and and of the nuclear spectral function (as depicted in Fig. 1.2 for lepton-nucleus deep inelastic inclusive scattering), by the following convolution integrals [70]

(1.6)
(1.7)

where is the nuclear spectral function which defines the joint probability of finding a nucleon in the nucleus with momentum and removal energy , is the mass of the nucleus, , , , and . The four vector momentum of the bound nucleon in the light-front coordinate frame is defined as , with , and .

1.2 The Nuclear European Muon Collaboration (EMC) Effect

Since the binding energy of the nucleus is very small compared to the energy scales in deep inelastic scattering, it was assumed that, except for nucleon Fermi motion, the nucleus acted as a collection of slowly moving weakly bound nucleons, with their internal properties unchanged compared to the free nucleon case. Therefore the following ratio for the per nucleon inelastic structure functions to the inelastic structure functions and of a free proton and neutron

(1.8)

was expected to rise for x 0.2, and be about 1.2 -1.3 for x = 0.65, as it is shown in Fig. 1.3, in which Fermi motion corrections were included. Hence, it was expected that the nuclear deep inelastic inclusive scattering cross section will be completely defined by partonic distributions of free nucleons, with the nucleon Fermi motion as the only nuclear effect. It was also predicted that Eqs. (1.6) and (1.7), would give the same results for all nuclei. In summary, it was predicted that the nuclear cross section would be the sum of the cross sections of the number of nucleons inside the target[38, 76, 108].

Figure 1.3: Theoretical predictions for the Fermi motion corrections of the bound nucleon structure functions for iron. Figure from [38].

The above assumption was demonstrated to be wrong when the European Muon Collaboration (EMC) experiment discovered that in the muon-nucleus DIS, the ratio of the scattering cross section from nuclei to the deuteron, which is close to the ratio R in Eq. (1.8), was in strong disagreement with the predictions of Fig. 1.3. In the initial experiments, the ratio of the structure functions of iron nucleus and deuteron (F(Fe)/F(D)) was experimentally found to be decreasing and substantially different than unity in the region (see Fig. 1.4). The interpretation of the experimental results was that the inelastic structure functions of nucleons measured in nuclei are different from those of quasi-free nucleons in the deuteron [38, 42]. It was also found that this difference was growing with the mass number A of the nucleus, which indicated that the nucleon structure modification is proportional to the nuclear density. The modification of the quark momentum distribution of bound nucleons in the nucleus, as compared to that of free nucleons, became known as the nuclear European Muon Collaboration ( EMC) effect.

Figure 1.4: EMC experimental results for the ratio of the structure functions of iron nucleus and deuterium (F(Fe)/F(D)). Figure from [38].

Recent measurements at the Jefferson lab (shown in Fig. 1.5 for Carbon) verified the nuclear EMC effect with unprecedented accuracy for a wide range of nuclei, confirming that the EMC ratio R= 2 / (A) ( where is the average cross section for isoscalar nucleon and is the cross section for deuteron) is below unit for all the targets studied [91].

Figure 1.5: Carbon EMC ratios for the highest settings ( quoted at x = 0.75). Uncertainties are the combined statistical and point-to-point systematic. The solid curve is the SLAC fit to the Carbon EMC ratio. Figure from [91].

1.3 Multinucleon short-range correlations in nuclei

The main theoretical picture that describes the bulk properties of nuclei, is that the nucleons are independent particles moving in an average or mean field generated by the remaining nucleons in the nucleus. As a result each nucleon is independent of the exact instantaneous position of all other nucleons. The simple mean field model has been successful in correctly predicting all nuclear magic numbers, as well as in describing a large amount of nuclear data [14, 19, 50].

Figure 1.6: 2N and 3N scaling regions for nucleus A. Figure from Ref.[84]

The nuclear shell model, based on the above described mean field picture of the nucleus, is valid for long range ( 2 fm) mutual separation of nucleons, which in the momentum space corresponds to the nucleon momentum being less that 250 MeV/ ([60]. The nuclear shell model however was found to break down for inter nucleon distances smaller than , where fm is the radius of the nucleon, on which two nucleons start to overlap and the notion of the mean field become invalid. Their dynamics are mainly defined by the NN interaction at short distances which is dominated by the tensor interaction ( fm) and repulsive core ( fm) [32, 60, 86]. Such configurations are generally referred to as 2N Short Range Correlations (SRCs) in the nucleus [48, 46, 89, 133, 99].

Theoretical analysis show that the nucleons belonging to such 2N SRCs have large (greater than ) relative momentum and low (smaller than ) center of mass momentum. There are also lower probability configurations such as three (3N) or multi-nucleon (MN) SRCs, a very important high density structures present in the ground state wave function of the nucleus, responsible for high momentum nucleons much above the Fermi momentum . Any experiment designed to access such MN SRCs must probe the bound nucleon in the nucleus at very large momenta. Lepton-nucleus scattering, at large values of the Bjorken parameter, , is the most appropriate experiment to prove such MN SRC nuclear structures.

The kinematic region for lepton-free nucleon scattering is 0 1, whereas for bound nucleon in a nucleus A is 0 A. It is expected that scattering from j-nucleon SRC will dominate at j-1 j[49]. If a lepton scatters off the nucleon from j-nucleon SRCs, then it is expected that the cross section ratio

(1.9)

where and are the inclusive lepton scattering cross sections of nucleus and respectively, will scale, that is to be constant. The scaling results from the dominance of MN SRCs in the high momentum component of the nuclear wave function. Hence, plateaus are expected in the ratio of the inclusive cross sections of heavy nucleus to light nuclei such as He, showing that the momentum distributions at high momenta have the same shape for all nuclei differing only by a scale factor [43, 78, 49].

Recent experimental studies of high energy and processes [78, 84, 103, 64, 74, 85, 90, 119] resulted in a significant progress in understanding the dynamics of 2N SRCs in nuclei. The series of electron-nucleus inclusive scattering experiments[78, 84, 103] have confirmed the prediction of the scaling for the ratios of inclusive cross sections of a nucleus to the deuteron (He) in the kinematic region (dominated by the scattering from the bound nucleons with momenta  MeV/) Within the 2N SRC model, these ratios allowed to extract the parameter which characterizes the probability of finding 2N SRC in the nucleus relative to the deuteron. Results of the above mentioned experiments with 1.5 GeV, are shown in Fig. 1.6.The cross section ratios in Fig. 1.6 scales initially in the region 1.5 x 2.0, which indicates dominance of 2N SRC in this region, and scales a second time for x 2.25, indicating dominance of 3N SRC.

1.3.1 Strong Correlation Between Nucleon-Nucleon Short Range Correlations and the EMC Effect

Since both the EMC effect and the 2N SRCs depend on the mass number A and on the nuclear density, it was predicted that they were strongly correlated [92]. The prediction was probed by experiments on which the correlation between the strength of the nuclear EMC effect and the strength of 2N SRCs was observed as it is shown in Fig. 1.7 [97].

Figure 1.7: The EMC slopes versus the SRC scale factors. Figure from Ref. [97]

The strength of the EMC effect for a nucleus A in Fig. 1.7 is represented by the slope of the EMC ratio (R) of the per-nucleon deep inelastic cross section of nucleus A relative to the deuteron, dR/dx, in the region 0.35 x 0.7 [92]; and the strength of the 2N SRCs is represented by the nuclear scale factor of the nucleus A relative to the deuteron a (A/d), which represents the probability of having 2N SRCs in the nucleus A.

The EMC-SRC correlation is a very important experimental result that provides new insight into the origin of the EMC effect. Since the SRC structure in the nucleus implies high momentum bound nucleons, it indicates that the EMC effect is only the result of the high momentum component of the nuclear wave function, so that the possible modification of the parton distributions in nucleons in the nucleus occurs only in nucleons belonging to SRCs [107, 108].

Hence, a further understanding of the dynamical origin of the observed EMC-SRC correlation requires a theoretical model for the nuclear spectral function, , that describes the 2N and 3N SRCs in a consistent way. The model will allow the calculation of the cross sections for lepton -nucleus scattering from Eqs. (1.5), (1.6), and (1.7). Thus, the main motivation of the research presented in this dissertation was to develop a self consistent theoretical model for calculation of the nuclear spectral functions in the domain of 2N and 3N short range correlations. One of the important requirements in developing such models was that the calculated spectral functions should include all recent findings that have been made in experimental and theoretical studies of SRCs in nuclei as those described in the following sections (1.3.2 and 1.3.3) of the present chapter.

1.3.2 Model for 2N and 3N short-range correlations in nuclei

Figure 1.8: 2N and 3N SRC models for nuclear spectral functions

Despite impressive recent progress in ab initio calculations of nuclear structure for the mean field (shell) model (see e.g. Ref.[118]), their relevance to the development of the spectral functions at large momenta and removal energies, where the dominance of SRCs is expected, is rather limited. Not only the absence of relativistic effects but also the impossibility of identifying the relevant nucleon-nucleon (NN) interaction potentials makes such a program unrealistic. One way for progress is to develop theoretical models that use the short-range NN correlation approach in the description of the high momentum part of the nuclear wave function (see, e.g., [36, 45, 52, 105, 109, 110, 116, 124, 128, 129]). In such an approach, it is possible to consider the empirical knowledge of SRCs acquired from different high energy scattering experiments thus reducing in some degree the theoretical uncertainty related to the description of high momentum nucleon in the nucleus.

The main goal of chapter 2 of the present dissertation is to develop a model for high momentum nuclear spectral function using the several phenomenological observations obtained in recent years in studies of the properties of two-nucleon SRCs[78, 84, 85, 90, 88, 103, 106, 130, 119, 131]. A model is first developed describing the nuclear spectral function at large momenta and missing energies dominated by 2N SRCs with their center of mass motion generated by the mean field of the residual nuclear system.

Even though some experiments have shown evidence for 3N SRCs [84], there are other experiments [103] that did not see such evidence. Considering the experimental ambiguity, a theoretical framework for calculating the contribution of 3N SRCs to the nuclear spectral function is developed, using a model in which such correlations are generated by two sequential 2N short range correlations as it is shown in Fig. 1.8. Hence, the phenomenological knowledge of the properties of 2N SRCs is sufficient to calculate both the 2N and 3N SRC contributions to the model of nuclear spectral function.

The model is assumed to be valid for nucleon momenta 1000 MeV/c, where is a momentum characteristic to the 2N SRC. It is sufficiently large that 2N SRCs can be factorized from residual mean field interaction. As a result the model will have limited validity in the transitional region of where the role of the long-range correlations are more relevant.

1.3.3 Phenomenology of two nucleon short-range correlations in nuclei

High energy semi-inclusive experiments[85, 90] probed for the first time the isospin composition of 2N SRCs, observing strong (by factor of 20) dominance of the SRCs in nuclei, as compared to the and correlations, for internal momentum range of  MeV/. The experimental results are understood by considering the dominance of the tensor forces in the NN interaction at the momentum range corresponding to the average nucleon separations of  fm [85, 82]. The tensor interaction projects the NN SRC part of the wave function to the isosinglet - relative angular momentum, , state, almost identical to the high momentum part of the -wave component of the deuteron wave function. As a result and components of the NN SRC are strongly suppressed since they are dominated by the central NN potential with relative angular momentum [33, 60, 86] .

On the basis of the above observation of the strong dominance of SRCs, it was predicted that single proton or neutron momentum distributions in the 2N SRC domain are inversely proportional to their relative fractions in nuclei [112, 104]. The prediction is in agreement with the results of variational Monte-Carlo calculation of momentum distributions of light nuclei[120] as well as for medium to heavy nuclei following the SRC model calculations of Ref.[124]. The recent finding of the dominance in heavy nuclei (up to Pb)[119] validates the universality of the above prediction for the whole spectrum of atomic nuclei. The inverse proportionality of the high momentum component to the relative fraction of the proton or neutron is important for asymmetric nuclei and they need to be included in the modeling of nuclear spectral functions in the 2N SRC region.

The dominance in the SRC region and its relation to the high momentum part of the deuteron wave function makes the studies of the deuteron structure at large internal momenta a very important part for the SRC studies in nuclei. In this respect, the recent experiments [96, 127] and planned new measurements [115] of high energy exclusive electro-disintegration of the deuteron opens up new possibilities in the extraction of the deuteron momentum distribution at very large momenta. The measured distributions can then be utilized in the calculation of the nuclear spectral functions in the multi-nucleon SRC region.

Finally, another progress relevant to the SRC studies was the extraction of the center of mass momentum distribution of 2N SRCs from the data on triple coincidence scattering in [79] and [87, 121] reactions. The Gaussian form and the width of the extracted distributions were in a good agreement with the predictions made in Ref. [52], which were based on the estimate of the mean kinetic energy of the NN pair in shell-model description of nuclei. Similar results have been also obtained within the correlated wave function method of Ref. [122].

1.4 Nuclear spectral function model

The above discussed phenomenology will provide the necessary empirical input for modeling nuclear spectral functions in the SRC region. The model for high momentum nuclear spectral function, developed in chapter 2 of this dissertation, have two regions determined by the range of momentum considered. For momenta below the Fermi momentum, , a mean field spectral function is constructed by using a nonrelativistic approach to estimate the ground state wave functions [80]. For momenta above 400 MeV, a relativistic multi-nucleon short range correlation model of the spectral function is obtained, which describes the high momentum and high missing energy of two and three nucleons in short range correlations (2N and 3N SRC), for symmetric and asymmetric nuclei.

Since the domain of multi-nucleon SRCs is characterized by the relativistic momenta of the probed nucleon, special care should be given to the treatment of relativistic effects. To identify the relativistic effects, in Sec.2, the nuclear spectral function is defined as a quantity which is extracted in the semi-exclusive high energy process whose scattering amplitude can be described through the covariant effective Feynman diagrams. The covariance here is important to consistently trace the relativistic effects related to the propagation of the bound nucleon. Then, the part of the covariant diagram which reproduces the nuclear spectral function is precisely identified. Two approaches are adopted for modeling the nuclear spectral function: virtual nucleon and light-front approximations, general features of which are described in Sec.2.1. Section 2.2 outlines the calculation of nuclear spectral functions using the effective Feynman diagrammatic method, identifying the diagrams corresponding to the mean field, 2N SRC with center of mass motion and 3N SRC contributions.In Secs.2.3 and 2.4, the detailed derivation of the nuclear spectral functions within the virtual nucleon and the light-front approximations are presented.

Chapters 3 and 4 are dedicated to the development of computational models to evaluate nuclear spectral functions, density matrices and momentum distributions for a wide range of light and heavy nuclei. The results of the spectral function models will be compared with ab initio, nonrelativistic quantum Monte Carlo calculations (QMC) (for A 11) [51, 67, 71, 120]. The values of and are also predicted, which represent the probability of having 2N and 3N SRCs in the nuclear ground state wave function, respectively.

Chapter 2 Multinucleon short-range correlation model for nuclear spectral functions: Theoretical framework

The definition of nuclear spectral functions used in the present dissertation is derived by identifying a nuclear “observable" which can be extracted from the cross section of the large momentum (nucleon mass) transfer semi-inclusive reaction in which the can be unambiguously identified as a struck nucleon carrying almost all the energy and momentum transferred to the nucleus by the probe . The reaction is specifically chosen to be semi-inclusive so that it allows, in the approximation in which no final state interactions are considered, to relate the missing momentum and energy of the reaction to the properties of bound nucleon in the nucleus. When those conditions are satisfied the extracted “observable", referred to as a nuclear spectral function, represents a joint probability of finding a bound nucleon in the nucleus with given missing momentum and removal energy .

The models of nuclear spectral functions developed in the present dissertation must be relativistic, since they will be used to describe bound nucleons with high momenta and high removal energies. The relativistic effects are accounted for by using effective Feynman diagrammatic approaches similar to those developed in Refs. [53, 61, 68, 81]. One problem associated with the relativistic domain is the existence of vacuum fluctuations that implies the existence of negative energy components which are not related to the probability amplitude of finding a nucleon with a given momentum in the nucleus, and therefore, are not components of the nuclear spectral function.

Chapter 2 is organized as follows. The two approaches to deal with vacuum fluctuations: virtual nucleon (VN) and light-front (LF) approximations are described in section 2.1. Section 2.2 outlines the modeling of nuclear spectral functions using the effective Feynman diagrammatic method, identifying the diagrams and the corresponding amplitudes which represent partial contributions to the total amplitude by nucleons in the nuclear mean field, in two nucleons short-range correlation with center of mass motion, and in three nucleons short-range correlation. The steps for the calculation of the models of the nuclear spectral function are also defined in section 2.2. Sections 2.3 and 2.4 include the detailed derivations of the models of nuclear spectral functions within the virtual nucleon and the light-front approximations respectively. Section 2.5 summarizes the results of the chapter.

2.1 Approaches to deal with vacuum fluctuations

The vacuum fluctuations are a purely relativistic phenomena associated with the existence of particles and antiparticles that can pop out from the vacuum and then disappear into it. The notion of the antiparticle was first proposed by Dirac in 1932 to explain the solutions of the Klein-Gordon equation with negative energies. The positron was predicted as the antiparticle of the electron with a positive charge and a negative energy.

Stuckelberg in 1941 and Feynman in 1948 proposed that a negative energy solution describes a particle which propagates backward in time, or equivalently a positive energy antiparticle propagating forward in time. This concept was incorporated in the Feynman diagrams, a very powerful method of calculation in quantum field theory. With the help of Feynman diagrams it is possible to show that, for certain time ordering of process, a pair particle- antiparticle may appear spontaneously from the vacuum, and unless there is some external energy carried by a probe, the pair will disappear back into it [40].

The models, developed in the present dissertation, for relativistic nuclear spectral functions are derived from an effective Feynman diagrammatic approach for calculation of the reactions (Fig.2.1) derived in Refs. [53, 61, 68]. In the approach the covariant Feynman scattering amplitude is expressed through the effective nuclear vertices, vertices which are related to the scattering of the probe with the bound nucleon, as well as vertices related to the final state of the reaction.

The nuclear vertices related with the bound nucleon can not be associated a priori with the single nucleon wave function of the nucleus, since they contain negative energy components which are related to the vacuum fluctuations rather than the probability amplitude of finding nucleon with given momentum in the nucleus.

Figure 2.1: Representation of the covariant Feynman amplitude through the sum of the time ordered amplitudes. Panel (b) corresponds to the scenario in which first, the bound nucleon is resolved in the nucleus which interacts with the incoming probe . In panel (c), initially, the incoming probe produces a , anti-nucleon and nucleon pair, with subsequent absorption of the anti-nucleon in the nucleus.

The problem of vacuum fluctuations is illustrated in the diagrammatic representation of the reaction shown in Fig. 2.1, in which the covariant diagram (a) is a sum of two non-covariant time ordered scattering diagrams (b) and (c). Here, for the calculation of the Lorentz invariant amplitude of Fig. 2.1(a), the Feynman diagrammatic rules (given in Ref. [68]) can be applied. However the nuclear spectral function can only be formulated for the diagram of Fig. 2.1(b), where the time ordering is such that it first exposes the nucleus as being composed of a bound nucleon and residual nucleus, followed by an interaction of the incoming probe off the bound nucleon. The other time ordering [Fig. 2.1(c)] presents a very different scenario of the scattering in which the probe produces a , anti-nucleon and nucleon pair with subsequent absorption of the anti-nucleon in the nucleus. The later is usually referred to as a -graph and is not related to the nuclear spectral function. It is worth noting that the -graph contribution is a purely relativistic effect and does not appear in the non-relativistic formulation of the nuclear spectral function. The -graph contribution however increases with an increase of the momentum of the bound nucleon [see e.g. Ref. [36]].

The above discussion indicates that while defining the nuclear spectral function is straightforward in the non-relativistic domain (no -graph contribution), its definition becomes increasingly ambiguous with an increase of the momentum of the bound nucleon. The ambiguity is reflected in the lack of uniqueness in defining the nuclear spectral function in the domain where it is expected to probe SRCs inside the nucleus.

In the present dissertation two approaches are considered to deal with the vacuum fluctuations, so that a unique definition of the nuclear spectral function from the covariant scattering amplitude is obtained. In the first approach, referred to as the virtual nucleon (VN) approximation, the -graph contribution is neglected [93, 98]. In the second approach, referred to as the light front (LF) approximation, the -graph contribution is kinematically suppressed [25, 36, 46, 66].

2.1.1 Virtual nucleon (VN) approximation approach

In the virtual nucleon (VN) approximation approach, the -graph contribution is neglected by considering only the positive energy pole for the bound nucleon propagator in the nucleus. The energy and momentum conservation in the VN approach requires the interacting nucleon to be virtual which renders certain ambiguity in treating the propagator of the bound nucleon. The ambiguity is solved by recovering the energy and momentum of the interacting nucleon from kinematic parameters of on-shell spectators [see Ref. [37] for general discussion of the spectator model of relativistic bound states].

The advantage of the VN approximation is that the spectral function is expressed through the nuclear wave function defined in the rest frame of the nucleus which in principle can be calculated using conventional NN potentials.

One shortcoming of the VN approximation is that while it satisfies the baryonic number conservation law, the momentum sum rule is not satisfied reflecting the virtual nature of the probed nucleon in the nucleus.

2.1.2 Light-front (LF) approximation approach

The light-front representation of the space-time was first proposed by Dirac [10] as a form of relativistic Hamiltonian dynamics. The three forms of Hamiltonian dynamics described by Dirac were: the instant form, the front form, and the point form. These forms differ in the hypersphere on which the fields are analyzed [62]. In the front form the hypersphere is a plane tangent to the light-cone, that is the three-dimensional surface in space-time formed by a plane wave frame advancing with the velocity of light, such a surface was called a front by Dirac.

The space-time coordinates in the light-front are defined as [62, 63]:
Lorentz Vectors: The contravariant four-vectors of position are written as

(2.1)

Its time-like and space-like components are related to the instant form by:

(2.2)

respectively, and referred to as the light-front time and light-front longitudinal position. The null plane is defined by x = 0, that is, this condition defines the hyperplane that is tangent to the light-cone. The initial boundary conditions for the dynamics in the light-front are defined on this hyperplane. The axis x is perpendicular to the plane x= 0. Therefore a displacement of such hyperplane for is analogous to the displacement of a plane in t = 0 to t of the four-dimensional space-time. With this analogy, x is recognized as the time in the light-front, or equivalently, as the light-front time . The contravariant four-vectors of momentum are written as

(2.3)

Its time-like and space-like components are related to the instant form by:

(2.4)

The scalar product between two 4-vectors is defined by:

(2.5)

if , then the following very useful relation is obtained:

(2.6)

where is the mass of the particle with momentum .
The four-dimensional phase space differential element is defined as

(2.7)

In the light-front (LF) approximation the nuclear spectral function is defined on the light front which corresponds to a reference frame in which the nucleus has infinite momentum. Weinberg [25] showed that in the infinite momentum frame all diagrams with negative energy, like the the -graph, are kinematically suppressed. Then, as a result, the invariant sum of the two light-cone time ordered amplitudes in Fig. 2.1 is equal to only the contribution from the graph of Fig. 2.1 (b). Therefore, the boost invariant LF nuclear spectral function defines the joint probability of finding a nucleon in the nucleus with given light-front momentum fraction, transverse momentum, and invariant mass [58].

It is worth noting that the LF approximation satisfies the baryonic conservation law, as well as the momentum sum rules, thus providing a better framework for studies of the effects associated with the nuclear medium modification of interacting particles.

The LF approach developed in the present dissertation is field-theoretical, that is the Feynman diagrams are constructed with effective interaction vertices and the spectral functions are extracted from the imaginary part of the covariant forward scattering nuclear amplitude. Another approach, in LF approximation, is the construction of the nuclear spectral function based on the relativistic Hamiltonian dynamics representing the interaction of fixed number on-mass shell constituents [132].

2.2 Diagrammatic approach for modeling nuclear spectral functions

The application of the Feynman diagrammatic rules of Ref. [68] to obtain the mathematical model of nuclear spectral functions starts by identifying the effective interaction vertices shown in Fig. 2.2, such that the imaginary part of the covariant forward scattering nuclear amplitude will reduce to the nuclear spectral function either in VN or LF approximations. The specific form of the vertices can be established by considering the amplitude of Fig. 2.1(b), taking into account the kinematics of the mean field, the 2N, and the 3N SRC scattering within VN and LF approximations, and with subsequent factorization of the scattering factors related to the external probe . As a result the vertices will be different for mean field, 2N, and 3N SRCs. They will also depend on the VN or the LF approximations used to calculate the scattering amplitude.

In applying the diagrammatic approach, the forward nuclear scattering amplitude can be expressed as a sum of the mean field and the multinucleon SRC contributions as presented in Fig. 2.2, with (a), (b), and (c) corresponding to the contributions from nucleons in the nuclear mean-field, and from the 2N and the 3N short-range correlations respectively:

(2.8)

where , , and correspond to the contributions from the diagrams of Fig. 2.2 (a)-2.2(c) respectively.

Since the mean field contribution is dominated by the momenta of interacting nucleon below the characteristic Fermi momentum, , it is valid to approximate the corresponding nuclear spectral function to the result following from nonrelativistic calculation. Hence, both the VN and the LF approximations are expected to give very close results.

Figure 2.2: Expansion of the nuclear spectral function into the contributions of the mean field (a), the 2N (b) and the 3N (c) SRCs. For each case the initial nuclear transition vertices are different, corresponding to transition of ; and for the mean field, the 2N and the 3N SRCs respectively. The NN (b) and NNN (c) labels identify the 2N and the 3N SRCs with effective vertices elaborated in the text.

For the 2N and 3N SRCs, the momenta of probed nucleon is in the range  MeV/ and the nonrelativistic approximation is increasingly invalid.

2.2.1 Covariant amplitude for a nucleon in the nuclear mean field

In the mean field approximation, the bound nucleon interacts with the nuclear mean field induced by the nuclear residual system. In such approximation, the nuclear spectral function corresponds to a configuration in which the residual nuclear system is identified as a coherent state with excitation energy in the order of tens of MeV.

Applying the effective Feynman rules [68] to the diagram of Fig. 2.2(a) corresponding to the mean field contribution of nuclear spectral function, the following covariant amplitude is obtained:

(2.9)

where and are the masses of the nucleon and of the nuclear residual system respectively, is the nuclear spin wave function, represents the covariant vertex of the transition, describes the propagation of the nuclear residual system in the intermediate state having an excitation . Following the effective Feynman rules [68], the propagator of the form and a factor have been assigned to the spectator . The label indicates that the cut diagram is estimated so that the residual nuclear system is on mass shell. The abbreviated notation , where are the Dirac matrices, has been used [40]

2.2.2 Covariant amplitude for two nucleons in short-range correlation

For two-nucleons in short-range correlation, it is assumed that the intermediate nuclear state consists of two correlated fast () nucleons and a slow () coherent nuclear residual system.

Applying the effective Feynman rules [68] to the diagram of Fig. 2.2(b) corresponding to the 2N SRC contribution of nuclear spectral function, the following covariant amplitude is obtained:

(2.10)

where is the mass of the 2N SRC system, now describes the transition of the nucleus A to the SRC and coherent nuclear residual state, while the vertex describes the short range interaction that generates two-nucleon correlation in the nuclear spectral function.

2.2.3 Covariant amplitude for three nucleons in short-range correlation

The nuclear spectral function that results from 3N short-range correlations is described in Fig. 2.2(c) in which the intermediate state consists of three fast () nucleons and a slow () coherent nuclear residual system.

The dynamics of the 3N SRCs allow more complex interactions than that of the 2N SRCs. One of the complexities is the irreducible three-nucleon forces that can not be described by the NN interaction only. Such interactions may contain inelastic transitions such as the interaction. Some studies demonstrated[82] that irreducible three-nucleon forces predominantly contribute at very large magnitudes of missing energy characteristic to the excitations  MeV/. Thus for nuclear spectral functions for which the missing energy does not exceed the resonance threshold , only the contributions of the interactions need to be considered.

Figure 2.3: Diagram corresponding to the 3N SRC contribution to the nuclear spectral functions.

In the two sequential NN short-range interaction scenario for the generation of 3N SRCs, the spectral function can be represented through the diagram of Fig. 2.3, where it is assumed that the contribution of the low momentum nuclear residual system is negligible. The assumption results from the fact that a much larger momenta are involved in the 3N SRCs as compared to the momenta in the 2N SRCs discussed in the previous section. As a result the effects due to the center of mass motion of the system can be safely neglected.

In the collinear approximation, the initial three collinear nucleons undergo two short-range NN interactions generating one nucleon with much larger momenta than the other two. It is assumed that the momentum fraction of the 3N SRCs carried by each initial nucleon is unity and that their total transverse momenta is neglected [see Eq. (2.113)]. Within the VN approximation, the collinear approximation assumes that the initial total momentum of the three nucleons is much smaller than -momenta characteristic to NN SRC- and therefore can be neglected (). The collinear approximation is commonly used in the calculation of the quark structure function of the nucleon in the valence quark region. Hence the calculations in the present dissertation for the LF approximation are analytically similar to the QCD calculation of the nucleon structure function.

Applying the effective Feynman rules [68] to the diagram of Fig. 2.3 corresponding to the 3N SRC contribution of the nuclear spectral function, the following covariant amplitude is obtained:

(2.11)

where “" labels the intermediate state of the nucleon 2 after the first short-range NN interaction, is the spin of the th nucleon and the is the same short range NN interaction vertex included in Eq. (2.10).

Note that there are several other 3N SRC diagrams which differ from that of Fig. 2.3 by the ordering of the two sequential NN short range interactions. In collinear approximation these diagrams result in the same analytic form both in VN and LF approximations (see, e.g., Ref. [125]), thus their contribution can be absorbed in the definition of the parameter [see Eq. (2.48)], which defines the contribution of the norm of the 3N SRCs to the total normalization of the nuclear spectral function.

2.2.4 Models calculation of nuclear spectral functions

The first step to calculate the nuclear spectral functions from the forward scattering amplitudes given by Eqs. (2.9)-(2.11) is to define the effective vertices which identify the bound nucleon in the mean field, in the 2N or in the 3N SRCs, as well as to define the poles at which the cut propagators of the intermediate states are estimated. Both will depend on the approach used to deal with vacuum fluctuations, that is virtual nucleon or light-front approximation.

After the vertex definition, the effective Feynman rules [68] are applied to the covariant forward scattering amplitudes corresponding to the mean field, the two-, or the three-nucleon SRC contributions (Fig. 2.2) separately. Within the VN or the LF approximation, the loop-integrals are calculated through the on-mass shell conditions of intermediate states, by integration through the positive poles of the cut propagators of the corresponding intermediate states. Finally, the numerators of the propagators are estimated by using sum rules representing the completeness relation for the wave functions of the intermediate states.

The following step is the definition of transition wave functions for the nucleus to nucleons and to nuclear residuals systems. Such definitions are formulated by the identification of the interaction diagrams for the bound states with the corresponding equations for the bound state wave function. For example in the non-relativistic limit, the interaction diagrams for the bound state, calculated based on the effective Feynman diagrammatic rules, are identified with the Lippmann-Schwinger equation [27, 30] in the non-relativistic limit. In the relativistic case, similar identifications are made with the Bethe-Salpeter type [12, 37] (for VN approximation) or the Weinberg type [25] (for LF approximation) equations for the relativistic bound state wave function.

Finally, the mathematical models for the SRC nuclear spectral functions are simplified by introducing effective momentum distributions for 2N SRC which are function of the deuteron momentum distribution, as well as approximate Gaussian distribution models for the center of mass motion of the 2N SRC. For the mean field nuclear spectral functions, a nonrelativistic approximation obtained from the conventional mean field calculations for a single nucleon is applied.

2.3 Nuclear spectral function in virtual nucleon approximation

The main assumptions of the model for nuclear spectral function in virtual nucleon approximation are that the nucleus is in the laboratory frame, the interacting bound nucleon is described as a virtual particle, and the spectators are put on their mass-shells.

The nuclear spectral function, , in virtual nucleon approximation is defined as the joint probability of finding a nucleon in the nucleus with momentum and removal energy . The conventional definition of the removal energy is

(2.12)

where and are the energy and the mass of the residual nuclear system respectively, and the nonrelativistic expression for the kinetic energy of system is subtracted. However, in practice the kinetic energy of the system depends on the mean field, the 2N-SRC or the 3N- SRC picture of the nuclear wave function, hence the corresponding kinetic energy will be accordingly defined for each particular case.

The following normalization condition for the nuclear spectral function in virtual nucleon approximation is defined with base on the conservation of baryonic number of the nucleus in hadron-nucleus scattering[41]:

(2.13)

where is the ratio of the flux factors of the (external probe)-(bound nucleon) and (external probe)-(nucleus) systems, which in the high momentum limit of the probe (hadron or virtual photon) yields

(2.14)

Here, and are the light-front longitudinal momenta of the bound nucleon and of the nucleus respectively, is the energy of the bound nucleon and the direction is defined opposite to the direction of the incoming probe.

Following the decomposition of Fig. 2.2, the mean field, the 2N, and the 3N SRC contributions to the nuclear spectral function are separately considered. In the VN approximation, the cut diagrams of Figs. 2.2 and 2.3 will be evaluated at the positive energy poles of the spectator residual system. For the mean field contribution, it corresponds to the positive energy pole of the coherent nuclear residual system. For the 2N SRC contribution, they correspond to the positive energy poles of the correlated nucleon and the nuclear residual system, whereas for the 3N SRC contribution, they correspond to the positive energy poles of the two correlated nucleons.

2.3.1 Nuclear spectral function in virtual nucleon approximation for a nucleon in the nuclear mean field

In the mean-field approximation [Fig. 2.2(a)] the missing momentum and the missing energy characterizes the total momentum and the excitation energy of the residual nuclear system. In the nuclear shell model, also defines the energy needed to remove the nucleon from a particular nuclear shell. For such a situation, the energy and the momentum of the interacting nucleon can be recovered from the kinematic parameters of the on-shell spectator.

The calculation of the nuclear spectral function in virtual nucleon approximation for a nucleon in the mean field of the residual nuclear system, starts with the following definition of the effective vertex in the mean field covariant amplitude Eq.(2.9)

(2.15)

where is the characteristic energy of the given nuclear shell, and the delta functions represent the momentum and the energy conservation in the vertex. The creation, (, and the annihilation, , operators in Dirac space are defined in such a way that they obey the following relation:

(2.16)

where is the spinor of the virtual nucleon.

Since in the VN approximation the vacuum fluctuations are neglected, the integral in Eq. (2.9), needs only to be calculated through the positive energy pole of the propagator of the on-shell residual nuclear system. Since this pole is displaced slightly below the real axis by the infinitesimal factor , the integral by is obtained by applying the Cauchy’s residue theorem to the lowest-half complex plane semicircle enclosing the displaced pole in the positive sense (clockwise direction), namely

(2.17)

where .

Let be the the spin wave function of the on-shell residual nuclear system, then the numerator of the propagator in Eq. (2.9), can be expressed by the following sum rule representing the completeness relation for the wave function:

(2.18)

It is important to note that in the relativistic treatment, the spin wave functions are momentum dependent as it is indicated in the argument of . Such a momentum dependence is also accounted for the spin wave function of other particles discussed in the present dissertation.

The definition of the vertex (2.15) follows the convention (see, e.g., Ref. [62]) for which the annihilation operator projects the nucleon propagator to the positive energy state, and the nuclear transition vertex produces the Fock component of the nuclear wave function. Note that the above definition is different from the conventional definition (see e.g. Ref. [128]) in which the annihilation operator acts over the nuclear wave function to produce nucleon-hole states. However the final results in both approaches are similar in the nonrelativistic limit.

Hence, using the above convention and from Eqs. (2.16)- (2.18), the single nucleon wave function, , for the given nuclear shell is defined as

(2.19)

Inserting the above wave function and the mean field vertex (2.15) into Eq.(2.9), and summing over all possible nuclear shells, , and spin projection, and , gives the following expression for the nuclear spectral function in virtual nucleon approximation for a nucleon in the mean field

(2.20)

which defines the joint probability of finding a nucleon in the mean field of the nucleus with momentum and removal energy .

Integration of Eq. (2.20) over and through the delta function, , yields

(2.21)

In order to obtain numerical estimates of the nuclear spectral function (2.21), it is important to consider that in the mean field approximation, the substantial strength of the wave function comes from the momentum range . Hence, the nonrelativistic approximation is valid, so that the mean field wave function (2.19) can be approximated by the nonrelativistic wave function obtained from the conventional mean field calculations of the single nucleon wave functions.

Additionally, in the nonrelativistic limit , and, in Eq.(2.13), the part does not contribute to the integral, resulting in the condition for the nonrelativistic normalization:

(2.22)

where is the mean field contribution to the total normalization of the nuclear spectral function.

2.3.2 Nuclear spectral function in virtual nucleon approximation for two nucleons in short-range correlation

The calculation of the nuclear spectral function in virtual nucleon approximation for two nucleons in short-range correlation, starts with the definition of the removal energy for the 2N SRC in which the correlated NN pair has a total momentum , in the mean field of the residual nuclear system. The magnitude of is therefore defined as

(2.23)

where is the threshold energy needed to remove two nucleons from the nucleus, with an approximated value of . Furthermore, and are the nonrelativistic kinetic energies of the and the residual nuclear systems respectively, and is the relativistic kinetic energy of the correlated nucleon 2. The expression for follows from the fact that in the rest frame of the nucleus , as well as from the definition of the removal energy in Eq. (2.12).

The effective vertex in the 2N SRC covariant amplitude (2.10) is defined as

(2.24)

where the creation, , and the annihilation, , operators of the nucleon with four-momentum and spin satisfy the relations of Eq. (2.16). The delta functions represent the momentum and energy conservation in the vertex.

Similarly to the propagator integration described in section 2.3.1, the integrations by and in Eq. (2.10) through the positive energy poles of the propagators of the on-shell particle , and nuclear residual system respectively, yields